As a British citizen I can explain why we have kept our monarchy.
Although the monarchy is the defacto head of state (and of several other states) the monarch has very little actual power. Though they could appoint a Prime Minister they would face public outcry.
An intresting point is that the armed forces swear alligence to his/her majesty, meaning that the army is under control of the king/queen. The monarch can order the troops to stand down from military operations, or even tell them to march on Westminster should the situation require it.
I can explain why Britain is still a monarchy in two words. Oliver Cromwell.
More seriously though, I think the chances of Britain becoming a republic in the 20th century are pretty slim with a post 1900 PoD.